# The language of any constant-time Turing machine is regular

Suppose we have a Turing machine $M$ so that there is a constant $t$ such that the Turing machine always runs in time $t$ or less. Prove that the language of $M$ is regular.

This seems to be a pretty well-known fact, but despite my research, I cannot seem to find a proof of this anywhere. Does someone know of one?

• @YuvalFilmus It would have been easier to edit and fix it before you wrote an answer. Nov 11 '14 at 9:26

Hint: If the machine runs in time $t$ on all inputs then its decision only depends on the first $t$ symbols of the input.